Commutators and square-zero elements in Banach algebras (Q2794435)

The paper deals mainly with the question: When does every commutator of an algebra lie in the closed linear span of the square-zero elements of the same algebra? For answering this question, the author introduces the property \({\mathcal B}_{\mathrm{alg}}\):NEWLINENEWLINEAn algebra \(A\) over a commutative unital ring \(C\) is said to satisfy the condition \({\mathcal B}_{\mathrm{alg}}\) if for every bilinear map \(f:A\times A\to X\), where \(X\) is an arbitrary \(C\)-module, the condition NEWLINE\[NEWLINE\text{for all }x, y\in A, \quad xy=0\Longrightarrow f(x, y)=0\tag{1}NEWLINE\]NEWLINE implies the condition NEWLINE\[NEWLINEf(xy, z)=f(x, yz)\text{ for all }x, y, z\in A.\tag{2}NEWLINE\]NEWLINE It is shown that if \(A\) is an algebra which satisfies the property \({\mathcal B}_{\mathrm{alg}}\) and also the property that \(A^2=A\), then every commutator in \(A\) is the sum of square-zero elements. In the case of an essential Banach algebra without the condition \(A^2=A\), every commutator lies in the closed linear span of square-zero elements.NEWLINENEWLINEIn Chapter 4, a description of all commutators as finite sums of square-zero elements is given in the case of algebras of square matrices with elements from a unital algebra. Using the results obtained for commutators and square-zero elements, some properties of the elements from the center of a semiprime (Banach) algebra are provided.